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This was a 'didn't finish' for me.

11d Lacking class, like New Yorker's behaviour (3-1)

I assume this is 'non-u' but can parse the New Yorker's behaviour bit.

10d Record lasting lifetime, but only the covers (10).

the non-u refers to US spelling of behavio(u)r

any crossing letters to help with the other?

Chronicle, record.
Chronic,lasting and outmost letters (only the covers) of Lifetime.

Thanks Culmer.
For 10d I've got C?????C?E

SB, our posts crossed...

Thanks Malone.